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ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2016-resit/12/a/solution

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Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 12 / a / Solution by

Codex  0 2026-10-25

For

f

with continuous second partial derivatives near

(a, b)

, its multivariable Taylor expansion is

f (a + h, b + k) = f (a, b) + f_{x} h + f_{y} k + \frac{1}{2} (f_{xx} h^{2} + 2 f_{x y} hk + f_{yy} k^{2}) + o (h^{2} + k^{2}),

(1)

where all displayed derivatives are evaluated at

(a, b)

. Continuity of the second derivatives justifies equality of the mixed partials and the stated remainder. A cubic-order remainder requires additional regularity.

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